Multifractal returns and hierarchical portfolio theory

We extend and test empirically the multifractal model of asset returns based on a multiplicative cascade of volatilities from large to small time scales. Inspired by an analogy between price dynamics and hydrodynamic turbulence, it models the time scale dependence of the probability distribution of returns in terms of a superposition of Gaussian laws, with a log-normal distribution of the Gaussian variances. This multifractal description of asset fluctuations is generalized into a multivariate framework to account simultaneously for correlations across time scales and between a basket of assets. The reported empirical results show that this extension is pertinent for financial modelling. Two sources of departure from normality are discussed: at large time scales, the distinction between discretely and continuously discounted returns leads to the usual log-normal deviation from normality; at small time scales, the multiplicative cascade process leads to multifractality and strong deviations from normality. By perturbation expansions of the cumulants of the distribution of returns, we are able to quantify precisely the interplay and crossover between these two mechanisms. The second part of the paper applies this theory to portfolio optimization. Our multiscale description allows us to characterize the portfolio return distribution at all time scales simultaneously. The portfolio composition is predicted to change with the investment time horizon (i.e. the time scale) in a way that can be fully determined once an adequate measure of risk is chosen. We discuss the use of the fourth-order cumulant and of utility functions. While the portfolio volatility can be optimized in some cases for all time horizons, the kurtosis and higher normalized cumulants cannot be simultaneously optimized. For a fixed investment horizon, we study in detail the influence of the number of rebalancing of the portfolio. For the large risks quantified by the cumulants of order larger than two, the number of periods has a non-trivial influence, in contrast with Tobin's result valid in the mean-variance framework. This theory provides a fundamental framework for the conflicting optimization involved in the different time horizons and quantifies systematically the trade-offs for an optimal inter-temporal portfolio optimization.